Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin (2017)

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S*^3*. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of uqc as a POVM that one recognizes to be a 3-manifold M*^*3. E. g., the d-dimensional POVMs defined from subgroups of finite index of the modular group PSL(2,ℤ) correspond to d-fold M*^*3– coverings over the trefoil knot. In this paper, one also investigates quantum information on a few \lq universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on SnapPy. Further connections between POVMs based uqc and M*^*3‘s obtained from Dehn fillings are explored.